• The complement of an event A contains all outcomes that are not in A. The union {A or B} of events A and B contains all outcomes in A, in B, and in both A and B. The intersection {A and B} contains all outcomes that are in both A and B, but not outcomes in A alone or B alone.
• The conditional probability P(B | A) of an event B, given an event A, is defined by
when P(A) > 0. In practice, conditional probabilities are most often found from directly available information.
275
• The essential general rules of elementary probability are
Legitimate values: 0 ≤ P(A) ≤ 1 for any event A
Total probability 1: P(S) = 1
Complement rule: P() = 1 − P(A)
Addition rule: P(A or B) = P(A) + P(B) − P(A and B)
Multiplication rule: P(A and B) = P(A)P(B | A)
• If A and B are disjoint, then P(A and B) = 0. The general addition rule for unions then becomes the special addition rule, P(A or B) = P(A) + P(B).
• A and B are independent when P(B | A) = P(B). The multiplication rule for intersections then becomes P(A and B) = P(A)P(B).
• In problems with several stages, draw a tree diagram to organize use of the multiplication and addition rules.